The problem requires calculating the number of walks of length $\ell$ between distinct vertices in the cycle graph $C_4$, where $\ell$ is given as the sum of the last two digits of your student number. Additionally, the hint suggests using the adjacency matrix of $C_4$.

Step-by-Step Solution Outline

1. Understanding the Graph $C_4$:
   The cycle graph $C_4$ has 4 vertices (labeled $v_1, v_2, v_3, v_4$) connected in a cycle: $v_1 \leftrightarrow v_2 \leftrightarrow v_3 \leftrightarrow v_4 \leftrightarrow v_1$
   The adjacency matrix $A$ of $C_4$ is symmetric and represents the connections (1 for edges, 0 otherwise):

   0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}$$ The matrix $$A$$ describes the neighbors of each vertex.

2. Walks of Length $\ell$ Using the Adjacency Matrix:
   The number of walks of length $\ell$ between two vertices in a graph is given by the entries of the matrix $A^\ell$, where $A^\ell$ is the adjacency matrix raised to the power $\ell$.

3. Steps to Solve:

   • Compute the powers $A^2, A^3, \dots, A^\ell$ using matrix multiplication.
   • Extract the appropriate entries of $A^\ell$ to count the number of walks between distinct vertices.

4. Specific Value of $\ell$:
   You need to determine the specific value of $\ell$ by summing the last two digits of your student number.

If you provide the value of $\ell$, I can compute the walks explicitly using the matrix powers.

Would you like me to calculate this for a specific value of $\ell$? Let me know your student number's last two digits, and I will proceed with the computation.

Related Questions:

1. How do you compute powers of an adjacency matrix for a general graph?
2. Why does the adjacency matrix provide information about walks in a graph?
3. How can eigenvalues of $A$ help compute powers of the adjacency matrix efficiently?
4. What is the structure of a cycle graph $C_n$ for any $n$?
5. What are the properties of walks versus paths in a graph?

Tip: For symmetric graphs like $C_n$, powers of the adjacency matrix have repeating patterns due to symmetry. This simplifies computations significantly.